There are a couple of ways to approach this. Initially, I would say that the sentence is not entirely wrong. I think that it can be refined and based on the different versions offered, you might be better suited to select which one captures the meaning you want. In the end, language is subjective in terms of the selection of words and phrases to convey what is in our mind, so I think that being able to select from an array of options might allow you to be able to fully capture what is present in your mind's eye. One thing you might want to start adopting is eliminating "a lot" in your writing. Changing that to, "much" or "a great deal of" each time you wish to you use "a lot" will help out "a lot" (Sorry, bad joke.) It will help out a great deal in clarity of your writing. The new sentence, then, could read, "Hurricanes can cause a great deal of damage because of high winds, rain, and storm surge." I would also suggest that the last noun is changed to a plural one, as in "storm surges" because hurricanes bring out more than one storm and the previous nouns have been pluralized ('rain" is more collective, than anything else.) The other thought I had was placing the more vivid parts of the sentence in front to grab the reader: "As a result of high winds, rain, and storm surges, hurricanes can cause a great deal of damage." These and the other options provided might be able to help bring out more richness in writing.
Besides avoiding the informal a lot that writing handbooks suggest not be used in college writing, you may wish to begin with the subordinate clause since doing so will place the emphasis on the more important idea:
Because of the high winds, rain, and storm surges that accompany them, hurricanes can cause great damage.
(The phrase that accompany them was added for clarification.)
Hurricanes can cause a lot of damage (or "great destruction" if you want to avoid using "a lot) because of high winds, heavy rains, and storm surges.